Topology of Algebraic Varieties

In many examples of moduli stacks which come equipped with a notion of stable points, one tests stability by considering "iso-trivial one parameter degenerations" of a point in the stack. To such a degeneration one can often associate a real number which measures "how destabilizing" it is, and in these situations one can ask the question of whether there is a "maximal destabilizing" or "canonically destabilizing" degeneration of a given unstable point. I will discuss a framework for formulating and discussing this question which generalizes several commonly studied examples: geometric invariant theory, the moduli of bundles on a smooth curve, the moduli of Bridgeland-semistable complexes on a smooth projective variety, the moduli of \(K\)-semistable varieties. The key construction in this story may be of independent interest: it assigns to any point in an algebraic stack a topological space parameterizing all possible iso-trivial degenerations of that point. When the stack is \(BG\) for a reductive \(G\), this recovers the spherical building of \(G\), and when the stack is \(X/T\) for a toric variety \(X\), this recovers the fan of \(X\).